Please only use definitions, propositions, theorems given in the book ''Differential Topology'' by Guillemin and Pollack !And if you refer to a proposition, theorem or exercise in the book please refer to them by chapter and section!Also please explain each step, even if it seems trivial to you! Prove that two maps of the circle S¹ into itself are homotopic if and onlyif they have the same degree. This is a special case of a remarkabletheorem of Hopf, which we will prove later. [HINT: If go, g₁ : R¹ → R¹both satisfy g(t + 1) = g(t) + 2nq, then so do all the maps g, = $8₁+ (1 - s)go.]

Question: Prove that two maps of the circle into itself are homotopic if and only if they have the…

Prove that two maps of the circle S¹ into itself are homotopic if and only if they have the same degree. This is a special case of a remarkable theorem of Hopf, which we will prove later. [HINT: If go, g₁ : R¹ R¹ both satisfy g(t + 1) = g(t) + 2лq, then so do all the maps g, = $8₁ + (1 - s)go.]

Prove that two maps of the circle S¹ into itself are homotopic if and only if they have the same degree. This is a special case of a remarkable theorem of Hopf, which we will prove later. [HINT: If go, g₁ : R¹ R¹ both satisfy g(t + 1) = g(t) + 2лq, then so do all the maps g, = $8₁ + (1 - s)go.]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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